Math’s Magic Square

This tutorial which I’m going to share today will be completely isolated from our other topics, and If you’ve ever got stuck on filling numbers in cubical boxes or squeezed while playing sudoku then this post is for you 


 Yes..! this post will surely blow your mind.

MAGIC SQUARE

Let’s have a look at the following question.

Q. A man has 25 cows and each cow is marked a number,  from 1 to 25. The number represents amount of milk produced by that cow. How can these cows be divided among 5 persons, such that all gets an equal number of cows producing an equal amount of milk?

Hint:  
( means 15th cow is producing 15 litres of milk)
we have to arrange the cows in such Manner that the number of cows if added it would be equal to the number of litres of milk.


Sol. This type of question can be solved by Magic Square Method which works on the concept of south-east direction.
confuse…!??
Don’t worry, read this till end…

Magic Squares’ is a term given to squares which are filled with consecutive integers and the total of whose rows, columns and diagonals is always the same. When the numbers in any row, column or diagonal are added up they reveal the same total.
A sample magic square is given below. It is a three-by-three grid and you will find that the total of all rows, columns and diagonals is 15. Since there are 9 squares in the grid, we have used numbers from 1 to 9.
math games
We can verify the various totals:
Row 1 : 4 + 3 + 8 = 15
Row 2 : 9 + 5 + 1 = 15
Row 3 : 2 + 7 + 6 = 15
Column 1 : 4 + 9 + 2 = 15
Column 2 : 3 + 5 + 7 = 15
Column 3 : 8 + 1 + 6 = 15
Diagonal 1 : 4 + 5 + 6 = 15
Diagonal 2 : 2 + 5 + 8 = 15
similarly we can arrange any magic box, And thus we can form a 5×5 grid for given question. This is a five by-five grid and the total of all sides will add up to 65. Since, there are 25 squares in the grid, we have used numbers from 1 to 25.
magic Square
You may verify the total of the rows, columns and diagonals. They will all add up to 65.

  • RULES of forming such squares :

(1) Always put the number 1 in the centre-most square of the last column.
(2) After inserting a number in a square move to the square in the south-east direction and fill it with the next number
(3) If the square in the south-east direction cannot be filled, then move to the square in the west and fill it with the next number.
(4) When you have filled a number in the last square of the grid, fill the next number in the square to its west.

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Explanation
A key of directions is given below for your convenience:
Math magic
(a) First, we follow rule 1 and place the first number 1 in the centre-most square of the last column
maths formula
(b) Next, we move to the south-east direction from this square. However, there is nothing in the south-east direction and hence we have to create an imaginary square in the south east direction. As per the rules the digit 2 will come in the imaginary square. However, you cannot put anything in an imaginary square and hence the number 2 will be written in a square farthest from it. Hence,
the number 2 will come as shown below.

(c) Now from the square where we have written the number 2 we come to the south-east and fill the number 3.
magic square
(d) Next, from 3 we move to the south-east and create an imaginary square. We fill the number 4 in the square farthest from this imaginary square. Refer to the figure given below:
math ticks

(e) From 4 we come to the south-east and fill the number 5 in the square to its south-east.

(f) Next, from 5 we have to come to the south-east to fill the number 6. But, we realise that the square in the south-east of 5 is already occupied by the digit 1. In this case we will follow the Rule 3 as mentioned above which
reads:
math tricks
If the square in the south-east direction cannot be filled, then move to the square in the west and fill it with the next number
Hence, we will fill the number 6 in the square which lies to the ‘west’ of 5. Refer to the diagram given below:

math tricks

Hence the topic ended but you can try this Maths magic square box method for further large numbers i.e. 49, 81 etc.

Hope you enjoyed and found worthy this post, How you feel with this tutorial, let me know by comment. I would like to hear from you.

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